613 = Primes: prime number, first number of prime triple (p, p + 4, p + 6), middle number of sexy prime triple (p − 6, p, p + 6). Geometrical numbers: Centered square number with 18 per side, circular number of 21 with a square grid and 27 using a triangular grid. Also 17-gonal. Hypotenuse of a right triangle with integral sides, these being 35 and 612. Partitioning: 613 partitions of 47 into non-factor primes, 613 non-squashing partitions into distinct parts of the number 54. Squares: Sum of squares of two consecutive integers, 17 and 18. Additional properties: a lucky number.

691 prime number, (negative) numerator of the Bernoulli numberB12 = -691/2730. Ramanujan's tau function τ and the divisor function σ11 are related by the remarkable congruence τ(n) ≡ σ11(n) (mod 691). In number theory, 691 is a "marker" (similar to the radioactive markers in biology): whenever it appears in a computation, one can be sure that Bernoulli numbers are involved.

Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing

Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the m

1.
This thermometer is indicating a negative Fahrenheit temperature (−4°F).

100 (number)
–
100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well

Factorization
–
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic buil

1.
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.

Divisor
–
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the sta

Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis fo

Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals co

1.
Entrance to section LII (52) of the Colosseum, with numerals still visible

4.
An inscription on Admiralty Arch, London. The number is 1910, for which MCMX would be more usual

Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers

Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less com

Quaternary numeral system
–
Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of bi

Quinary
–
Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today,

Senary
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor

Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decima

Duodecimal
–
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten an

Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a

Vigesimal
–
The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters

Base 36
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor

Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chai

2.
Natural numbers can be used for counting (one apple, two apples, three apples, …)

Composite number
–
A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number be

1.
Overview

Abundant number
–
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number and its proper divisors are 1,2,3,4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance, the number 12 has an abundance of 4

1.
Overview

Pronic number
–
A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers, however, the rectangular number name has also been applied to the composite numbers. The first few numbers are,0,

1.
Overview

NASCAR
–
The National Association for Stock Car Auto Racing is an American family-owned and operated business venture that sanctions and governs multiple auto-racing sports events. Bill France Sr. founded the company in 1948 and his grandson Brian France became its CEO in 2003, NASCAR is motorsports preeminent stock-car racing organization. The three larges

Coca-Cola 600
–
The Coca-Cola 600 is an annual 600-mile Monster Energy NASCAR Cup Series points race held at the Charlotte Motor Speedway in Concord, North Carolina during Memorial Day weekend. The event, when first held in 1960, became the first race to be held at the Charlotte Motor Speedway, run since 1960, it is the longest race on NASCARs schedule at 600 mile

1.
Coca-Cola 600

Fiat 600
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The Fiat 600 is a city car produced by the Italian manufacturer Fiat from 1955 to 1969. Measuring only 3.22 m long, it was the first rear-engined Fiat, the total number produced from 1955 to 1969 at the Mirafiori plant in Turin was 2,695,197. Codenamed Progetto 100, the Fiat 600 mirrored the layout of the Volkswagen Beetle, a total of 5 prototypes

Latin language
–
Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages

Cohort (military unit)
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A cohort was the standard tactical unit of a Roman legion and was composed of 360 soldiers. A Cohort is considered to be the equivalent of a military battalion. The Cohort unit replaced the system following the reforms traditionally attributed to Gaius Marius in 107 BC. Until the middle of the first century AD,10 cohorts made up a Roman Legion, dur

Phoenix, AZ
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Phoenix is the capital and most populous city of the U. S. state of Arizona. Phoenix is the anchor of the Phoenix metropolitan area, also known as the Valley of the Sun, the metropolitan area is the 12th largest by population in the United States, with approximately 4.3 million people as of 2010. Settled in 1867 as a community near the confluence o

2.
The Phillip Darrell Duppa adobe house was built in 1870 and is the oldest known house in Phoenix. The homestead of "Lord" Darrell Duppa, an Englishman who is credited with naming Phoenix and Tempe as well as founding the town of New River.

3.
Aerial lithograph of Phoenix from 1885

4.
Central Avenue, Phoenix, 1908

Area code 480
–
North American area code 480 is a telephone area code in Arizona serving the eastern and northern portions of the Phoenix metropolitan area. It was created on April 1,1999 in a split of area code 602. Generally,480 is coextensive with the East Valley, while most of the West Valley is area code 623, original plans called for the 480 code to be an ov

Area code 623

New Hampshire
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New Hampshire is a state in the New England region of the northeastern United States. It is bordered by Massachusetts to the south, Vermont to the west, Maine and the Atlantic Ocean to the east, New Hampshire is the 5th smallest by land area and the 9th least populous of the 50 United States. Concord is the capital, while Manchester is the largest

Sphenic number
–
In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers. A sphenic number is a product pqr where p, q and this definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance,60 =22 ×3 ×5 has exactly 3 prime factors, the smallest sphenic number is

1.
Overview

Mertens function
–
In number theory, the Mertens function is defined for all positive integers n as M = ∑ k =1 n μ where μ is the Möbius function. The function is named in honour of Franz Mertens and this definition can be extended to positive real numbers as follows, M = M. Less formally, M is the count of square-free integers up to x that have a number of prime fac

1.
Mertens function to n=10,000

Fibonacci number
–
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n

2.
A tiling with squares whose side lengths are successive Fibonacci numbers

Markov number
–
The first few Markov numbers are 1,2,5,13,29,34,89,169,194,233,433,610,985,1325. Appearing as coordinates of the Markov triples, etc, there are infinitely many Markov numbers and Markov triples. There are two ways to obtain a new Markov triple from an old one. First, one may permute the 3 numbers x, y, z, second, if is a Markov triple then by Vieta

1.
The first levels of the Markov number tree

610 (telephone)
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A600 series connector is an obsolete three-pin connector with up to six conductors. It was for years the standard telephone service connector in Australia. As of 2008 a large installed base remained and a variety of connectors, adapters and equipment such as ADSL filters were still in production. It has no common or standard use, either within Aust

1.
600 series connectors

Australia
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Australia, officially the Commonwealth of Australia, is a country comprising the mainland of the Australian continent, the island of Tasmania and numerous smaller islands. It is the worlds sixth-largest country by total area, the neighbouring countries are Papua New Guinea, Indonesia and East Timor to the north, the Solomon Islands and Vanuatu to t

Area code 612
–
By geographical area, it is the smallest area code in the state of Minnesota. However, like every single metropolitan area code in the United States, the used to be much larger, accounting for the entire Twin Cities region. At the outset, Minnesota received two area codes,612 and 218, a 1947 map of the NANP showed the region defined as roughly the

1.
Metropolitan Minneapolis area codes with 612 in yellow.

Judaism
–
Judaism encompasses the religion, philosophy, culture and way of life of the Jewish people. Judaism is an ancient monotheistic Abrahamic religion, with the Torah as its text, and supplemental oral tradition represented by later texts such as the Midrash. Judaism is considered by religious Jews to be the expression of the relationship that God estab

Kabbalah
–
Kabbalah is an esoteric method, discipline, and school of thought that originated in Judaism. A traditional Kabbalist in Judaism is called a Mekubbal, Kabbalah is a set of esoteric teachings meant to explain the relationship between an unchanging, eternal, and mysterious Ein Sof and the mortal and finite universe. While it is used by some denominat

3.
Grave of Rabbi Akiva in Tiberias. He features in Hekhalot mystical literature, and as one of the four who entered the Pardes

Sefirah
–
The term is alternatively transliterated into English as Sefirot/Sefiroth, singular Sephirah/Sefirah etc. Alternative configurations of the sephirot are given by different schools in the development of Kabbalah. The tradition of enumerating 10 is stated in the Sefer Yetzirah, Ten sephirot of nothingness, ten and not nine, ten and not eleven. As alt

1.
The Yosher -Upright configuration of the 10 Sephirot, arranged into 3 columns

2.
Kabbalah relates the Sephirot and Indwelling Shechinah Presence to Male-Female Divine principles, represented in the union of Jewish marriage Below. In Medieval Kabbalah the task of man is Yichud-"Union" of Male and Female Divinity on High [citation needed]. In Lurianic Kabbalah man redeems exiled Sparks of Holiness of the Shechinah from material Kelipot

3.
Ten Sephirot on the east wall of the Cordoba Synagogue, Spain

613 mitzvot
–
The tradition that 613 commandments is the number of mitzvot in the Torah, began in the 3rd century CE, when Rabbi Simlai mentioned it in a sermon that is recorded in Talmud Makkot 23b. These principles of Biblical law are sometimes called connections or commandments and referred to collectively as the Law of Moses, Mosaic Law, Sinaitic Law, the wo

Torah
–
The Torah is the central reference of Judaism. It has a range of meanings and it can most specifically mean the first five books of the twenty-four books of the Tanakh, and it usually includes the rabbinic commentaries. In rabbinic literature the word Torah denotes both the five books and the Oral Torah, the Oral Torah consists of interpretations a

Madison Square Garden
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Madison Square Garden, often called MSG or simply The Garden, is a multi-purpose indoor arena in the New York City borough of Manhattan. Located in Midtown Manhattan between 7th and 8th Avenues from 31st to 33rd Streets, it is situated atop Pennsylvania Station. The Garden is used for basketball and ice hockey, as well as boxing, concerts, ice show

New York Knicks
–
The New York Knickerbockers, commonly referred to as the Knicks, are an American professional basketball team based in New York City. The Knicks compete in the National Basketball Association as a club of the Atlantic Division of the Eastern Conference. The team plays its games at Madison Square Garden, located in the borough of Manhattan. They are

1.
Lapchick was responsible for leading the Knicks during their early success. However, these ventures never culminated with a win in the NBA Finals.

2.
New York Knicks

3.
William 'Red' Holzman guided the Knicks to two championships during his tenure.

4.
Walt 'Clyde' Frazier

Red Holzman
–
William Red Holzman was an NBA basketball player and coach probably best known as the head coach of the New York Knicks from 1967 to 1982. Holzman helped lead the Knicks to two NBA Championships in 1970 and 1973, and was inducted into the Basketball Hall of Fame in 1986, in 1996, Holzman was named one of Top 10 Coaches in NBA History. Born in Brook

1.
Red Holzman in the 1970s

2.
Holzman in 1950, when playing for the Rochester Royals.

Padovan sequence
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The Padovan sequence is the sequence of integers P defined by the initial values P = P = P =1, and the recurrence relation P = P + P. The first few values of P are 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265. The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his

1.
Spiral of equilateral triangles with side lengths which follow the Padovan sequence.

Number of the Beast (numerology)
–
The Number of the Beast is a term in the Book of Revelation, of the New Testament, that is associated with the Beast of Revelation in chapter 13. In most manuscripts of the New Testament and in English translations of the Bible, in critical editions of the Greek text, such as the Novum Testamentum Graece, it is noted that 616 is a variant. In the o

666 (number)
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666 is the natural number following 665 and preceding 667. Six hundred and sixty-six is called the number of the Beast in chapter 13 of the Book of Revelation, of the New Testament,666 is the sum of the first 36 natural numbers, and thus it is a triangular number. Notice that 36 =15 +21,15 and 21 are also triangular numbers, in base 10,666 is a rep

Eisenstein prime
–
In mathematics, an Eisenstein prime is an Eisenstein integer z = a + b ω that is irreducible in the ring-theoretic sense, its only Eisenstein divisors are the units, a + bω itself and its associates. The associates and the conjugate of any Eisenstein prime are also prime. It follows that the absolute value squared of every Eisenstein prime is a pri

1.
Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3 n − 1. All others have an absolute value squared equal to a natural prime.

1.
Young diagrams associated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflection about the main diagonal of the square are conjugate partitions.

Triangular number

1.
The first six triangular numbers

Hexagonal number

1.
Contents

Centered triangular number

1.
This article has an unclear citation style. The references used may be made clearer with a different or consistent style of citation, footnoting, or external linking. (June 2014)

4.
This page from Athanasius Kircher 's Oedipus Aegyptiacus (1653) belongs to a treatise on magic squares and shows the Sigillum Iovis associated with Jupiter

Eight queens puzzle

Harmonic divisor number

1.
Overview

LIST OF IMAGES

1.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

2.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers

Negative number
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This thermometer is indicating a negative Fahrenheit temperature (−4°F).

3.
100 (number)
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100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number

4.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors

Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.

5.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number

6.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made

7.
Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used

Roman numerals
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Entrance to section LII (52) of the Colosseum, with numerals still visible
Roman numerals
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Numeral systems
Roman numerals
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A typical clock face with Roman numerals in Bad Salzdetfurth, Germany
Roman numerals
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An inscription on Admiralty Arch, London. The number is 1910, for which MCMX would be more usual

8.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra

Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
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Gottfried Leibniz
Binary number
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George Boole

9.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1

10.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits

11.
Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons

12.
Senary
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion

13.
Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three

14.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer

15.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus &#x2019, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly

16.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power

17.
Base 36
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion

18.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory

19.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N

Composite number
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Overview

20.
Abundant number
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In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number and its proper divisors are 1,2,3,4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance, the number 12 has an abundance of 4, for example. A number n for which the sum of divisors σ>2n, or, equivalently, the sum of proper divisors s>n. The first 28 abundant numbers are,12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,100,102,104,108,112,114,120, …. For example, the divisors of 24 are 1,2,3,4,6,8. Because 36 is more than 24, the number 24 is abundant and its abundance is 36 −24 =12. The smallest odd abundant number is 945, the smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5,7,11,13,17,19,23, and 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If A represents the smallest abundant number not divisible by the first k primes then for all ϵ >0 we have,2 − ϵ < ln ⁡ A <2 + ϵ for sufficiently large k, infinitely many even and odd abundant numbers exist. The set of abundant numbers has a natural density, marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480. Every multiple of a number is abundant. For example, every multiple of 6 is abundant because the divisors include 1, n/2, n/3, every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10, every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number which is not a number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The abundancy index of n is the ratio σ/n, distinct numbers n1, n2. with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 =12 corresponds to the first abundant number, if p = is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant

Abundant number
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Overview

21.
Pronic number
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A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers, however, the rectangular number name has also been applied to the composite numbers. The first few numbers are,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462 …. The nth pronic number is also the difference between the odd square 2 and the st centered hexagonal number. The sum of the reciprocals of the numbers is a telescoping series that sums to 1,1 =12 +16 +112 ⋯ = ∑ i =1 ∞1 i. The partial sum of the first n terms in this series is ∑ i =1 n 1 i = n n +1, the nth pronic number is the sum of the first n even integers. It follows that all numbers are even, and that 2 is the only prime pronic number. It is also the only number in the Fibonacci sequence. The number of entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a number is present in only one of the factors n or n+1. Thus a pronic number is squarefree if and only if n and n +1 are also squarefree, the number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n +1. If 25 is appended to the representation of any pronic number. This is because 2 =100 n 2 +100 n +25 =100 n +25

Pronic number
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Overview

22.
NASCAR
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The National Association for Stock Car Auto Racing is an American family-owned and operated business venture that sanctions and governs multiple auto-racing sports events. Bill France Sr. founded the company in 1948 and his grandson Brian France became its CEO in 2003, NASCAR is motorsports preeminent stock-car racing organization. The three largest racing-series sanctioned by this company are the Monster Energy NASCAR Cup Series, the Xfinity Series, the company also oversees NASCAR Local Racing, the Whelen Modified Tour, the Whelen All-American Series, and the NASCAR iRacing. com Series. NASCAR sanctions over 1,500 races at over 100 tracks in 39 of the 50 US states as well as in Canada. NASCAR has presented exhibition races at the Suzuka and Motegi circuits in Japan, the Autódromo Hermanos Rodríguez in Mexico, NASCAR has its official headquarters in Daytona Beach, Florida, and also maintains offices in the North Carolina cities of Charlotte, Concord, and Conover. Regional offices are located in New York City and Los Angeles, with offices in Mexico City. Owing to NASCARs Southern roots, all but a handful of NASCAR teams are based in North Carolina. NASCAR is second to the National Football League among professional sports franchises in terms of television viewers, internationally, its races are broadcast on television in over 150 countries. In 2004, NASCARs Director of Security stated that the company holds 17 of the Top 20 regularly attended single-day sporting events in the world, fortune 500 companies sponsor NASCAR more than any other motor sport, although this sponsorship has declined since the early-2000s. By the time the Bonneville Salt Flats became the location for pursuit of land speed records. Drivers raced on a 4. 1-mile course, consisting of a 1. 5–2. 0-mile stretch of beach as one straightaway, the two straights were connected by two tight, deeply rutted and sand covered turns at each end. Stock car racing in the United States has its origins in bootlegging during Prohibition, bootleggers needed to distribute their illicit products, and they typically used small, fast vehicles to better evade the police. Many of the drivers would modify their cars for speed and handling, as well as increased cargo capacity, the cars continued to improve, and by the late 1940s, races featuring these cars were being run for pride and profit. These races were popular entertainment in the rural Southern United States, most races in those days were of modified cars. Street vehicles were lightened and reinforced, mechanic William France Sr. moved to Daytona Beach, Florida, from Washington, D. C. in 1935 to escape the Great Depression. He was familiar with the history of the area from the speed record attempts. France entered the 1936 Daytona event, finishing fifth and he took over running the course in 1938. He promoted a few races before World War II, France had the notion that people would enjoy watching stock cars race

23.
Coca-Cola 600
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The Coca-Cola 600 is an annual 600-mile Monster Energy NASCAR Cup Series points race held at the Charlotte Motor Speedway in Concord, North Carolina during Memorial Day weekend. The event, when first held in 1960, became the first race to be held at the Charlotte Motor Speedway, run since 1960, it is the longest race on NASCARs schedule at 600 miles. It is also unique for the fact that the race changes drastically from start to finish and it starts around 6,20 PM and the track is bathed in sunlight for the first third of the race. The second third happens at dusk, and the third under the lights. Turner thought he could borrow money to build a $750,000 track with 45,000 permanent seats on his property in Cabarrus County. Afterward, he learned that a group led by Bruton Smith had an idea to build a track near Pineville. Smith and Turner formed an alliance to build the track, once the construction crew broke ground, they found a layer of granite under the topsoil, making the construction costly. The area for the first turn alone used $70,000 worth of dynamite, in the spring of 1960, Turner begged for a six-week postponement for the race after a snow storm delayed the pouring on concrete. With two weeks remaining until the race, the paving subcontractor threatened to leave the job site for lack of payment. To solve the problem, Turner and one of his friends threatened the paving subcontractor with a shotgun, the first event at the recently completed Charlotte Motor Speedway was held on June 19,1960. The event was started as an attempt by NASCAR to stage a Memorial Day weekend event to compete with the open-wheel Indianapolis 500 and it was not until 1974, however, that both races competed head-to-head on the same day. In fact, the first World 600 was not held on the Memorial Day weekend, the 2009 race, postponed by rain from its original May 24 date, was the first race to have run on Memorial Day itself. With the installation of lights in 1992, fans complained to management to have the race start later in the day because of the notorious North Carolina heat. They wanted to follow The Winstons popularity the previous week and switch the race to a finish to create cooler temperatures for spectators. The start time was moved several times throughout the 1990s. With the new starting time came new challenges, not only do race teams have to deal with the blistering Carolina heat, but the considerable temperature change at night make track conditions completely different. The nighttime portion of the race is lit with a system that uses parabolic reflectors so that dangerous glare that would otherwise be in the eyes is minimized. Experts disagreed over whether, for health and safety reasons, anyone should be allowed to race 1100 miles in one day, from 2005 to 2010, the issue became moot when the state of Indiana finally decided to go to daylight saving time

Coca-Cola 600
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Coca-Cola 600

24.
Fiat 600
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The Fiat 600 is a city car produced by the Italian manufacturer Fiat from 1955 to 1969. Measuring only 3.22 m long, it was the first rear-engined Fiat, the total number produced from 1955 to 1969 at the Mirafiori plant in Turin was 2,695,197. Codenamed Progetto 100, the Fiat 600 mirrored the layout of the Volkswagen Beetle, a total of 5 prototypes were built between 1952 and 1954, which all differed from one another. Chassis number 000001 with engine number 000002 is believed to be the remaining example. It was powered by an innovative single-cam V2-cylinder engine designed to simplify maintenance, at the official launch in 1955, FIAT engineer, Dante Giacosa declared that the aim had been to create something new, both in the interest of progress and simplification. This prototype, however, did not become the chosen design, the car had hydraulic drum brakes on all four wheels. All 600 models had 3-synchro 4-speed transaxles, all models of the 600 had generators with mechanical external regulators. The top speed ranged from 95 km/h empty with the 633 cc inline-four engine to 110 km/h with the 767 cc version, the car had good ventilation and defrosting systems. A year after its debut, in 1956, a version was introduced. It was a precursor of current multi-purpose vehicles, at the time when the millionth car was produced, the manufacturer reported it was producing the car at the then remarkable rate of 1,000 a day. As of 2011 there are only 65 left in the UK that are road legal, in Spain, the 600 model was made under the make of SEAT, from 1957 to 1973. Up to 797,319 SEAT600 were made, the Spanish car maker exported them to a number of countries worldwide. This car motorised Spain after the Spanish Civil War, the most interesting version produced between 1964 and 1967 by SEAT is the SEAT800, the sole four-door derivative of the 600 model which received a longer wheelbase. It was developed in-house by SEAT and produced exclusively by the Spanish car maker without any equivalent model in Fiats range, the Fiat 600 was also manufactured at Fiat Neckar in Germany between 1956 and 1967. Presented in a first time as Jagst 600, in 1960 with the release of Fiat 600D it became Jagst 770, the model was manufactured until the end of 1967, more than 172,000 copies. It was produced by the Zastava factory in Kragujevac, Serbia, from the early 1960s until 1985, during time it played a major role in motorisation of the country. The 600 was built as the Fiat 600 R by Sevel in Argentina from 1960 to 1982, with operations also taking place in Uruguay by Ayax S. A. At first, Someca S. A. built the 600 with rear-hinged doors, as a new plant was constructed in the Ferreyra, a suburb of Córdoba, the local parts content steadily increased

25.
Latin language
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole

26.
Cohort (military unit)
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A cohort was the standard tactical unit of a Roman legion and was composed of 360 soldiers. A Cohort is considered to be the equivalent of a military battalion. The Cohort unit replaced the system following the reforms traditionally attributed to Gaius Marius in 107 BC. Until the middle of the first century AD,10 cohorts made up a Roman Legion, during the 1st century AD, the command structure and make-up of the legions was formally laid down, in a form that would endure for centuries. The first cohort was now made up of five double-strength centuries totalling 800 men and this century was known as the primus pilii, and its centurion was known as the primus pilus. The Primus Pilus could be promoted to Praefectus Castrorum, or Camp Prefect, the Praefectus Castrorum would be in charge of the daily running of a legion. The other cohort consisted of approximately 480 men in six centuriae of 80 men, at various times prior to the reforms, a century might have meant a unit of 60 to 80. The cohort had no permanent commander, it is assumed that in combat, in order of seniority, the six centurions were titled hastatus posterior, hastatus prior, princeps posterior, princeps prior, pilus posterior and pilus prior. The legion at this time numbered about 5,400 men, including officers, engineers, auxiliary cohorts could be quinquagenaria or milliaria. The term was first used to refer to the bodyguard of a general during the Republic, later, cohors togata was a unit of the Praetorian guard in civilian dress tasked with duties within the pomerium. Cohortes urbanae, urban cohort, military police unit patrolling in the capital, cohortes vigilum, watchmen, unit of the police force which also was the fire brigade in the capital. Cohors Germanorum, the unit of Germani custodes corporis, furthermore, the Latin word cohors was used in a looser way to describe a rather large company of people. Auxiliaries List of Roman auxiliary regiments

27.
Phoenix, AZ
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Phoenix is the capital and most populous city of the U. S. state of Arizona. Phoenix is the anchor of the Phoenix metropolitan area, also known as the Valley of the Sun, the metropolitan area is the 12th largest by population in the United States, with approximately 4.3 million people as of 2010. Settled in 1867 as a community near the confluence of the Salt and Gila Rivers. Located in the reaches of the Sonoran Desert, Phoenix has a subtropical desert climate. Despite this, its canal system led to a farming community, many of the original crops remaining important parts of the Phoenix economy for decades, such as alfalfa, cotton, citrus. The city averaged a four percent annual growth rate over a 40-year period from the mid-1960s to the mid-2000s. This growth rate slowed during the Great Recession of 2007–09, and has rebounded slowly, Phoenix is the cultural center of the Valley of the Sun, as well as the entire state. For more than 2,000 years, the Hohokam people occupied the land that would become Phoenix, the Hohokam created roughly 135 miles of irrigation canals, making the desert land arable. Paths of these canals would later used for the modern Arizona Canal, Central Arizona Project Canal. The Hohokam also carried out trade with the nearby Anasazi, Mogollon and Sinagua. It is believed that between 1300 and 1450, periods of drought and severe floods led to the Hohokam civilizations abandonment of the area. After the departure of the Hohokam, groups of Akimel Oodham, Tohono Oodham and Maricopa tribes began to use the area, as well as segments of the Yavapai and Apache. The Oodham were offshoots of the Sobaipuri tribe, who in turn were thought to be the descendants of the formerly urbanized Hohokam and their crops included corn, beans, and squash for food, while cotton and tobacco were also cultivated. Mostly a peaceful group, they did together with the Maricopa for their mutual protection against incursions by both the Yuma and Apache tribes. The Tohono Oodham lived in the region as well, but their concentration was to the south. Living in small settlements, the Oodham were seasonal farmers who took advantage of the rains and they also hunted local game such as deer, rabbit, and javalina for meat. When the Mexican–American War ended in 1848, Mexico ceded its northern zone to the United States, the Phoenix area became part of the New Mexico Territory. In 1863 the mining town of Wickenburg was the first to be established in what is now Maricopa County, at the time Maricopa County had not yet been incorporated, the land was within Yavapai County, which included the major town of Prescott to the north of Wickenburg

28.
Area code 480
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North American area code 480 is a telephone area code in Arizona serving the eastern and northern portions of the Phoenix metropolitan area. It was created on April 1,1999 in a split of area code 602. Generally,480 is coextensive with the East Valley, while most of the West Valley is area code 623, original plans called for the 480 code to be an overlay code for the entire Phoenix metro area. However, overlays were a new concept at the time, and met considerable resistance due to potential geographic ambiguity caused by overlay codes. A three-way geographic split was chosen instead, with 480 being assigned to the East Valley, also in area code 480 are pagers for the Iridium satellite telephone service. As of October 2016, area code 480 is expected to reach exhaustion by the quarter of 2020, however, as of February 2016. NANPA Area Code Map of Arizona

Area code 480

29.
New Hampshire
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New Hampshire is a state in the New England region of the northeastern United States. It is bordered by Massachusetts to the south, Vermont to the west, Maine and the Atlantic Ocean to the east, New Hampshire is the 5th smallest by land area and the 9th least populous of the 50 United States. Concord is the capital, while Manchester is the largest city in the state and in northern New England, including Vermont. It has no sales tax, nor is personal income taxed at either the state or local level. The New Hampshire primary is the first primary in the U. S. presidential election cycle and its license plates carry the state motto, Live Free or Die. The states nickname, The Granite State, refers to its extensive granite formations, the state was named after the southern English county of Hampshire by Captain John Mason. New Hampshire is part of the New England region and it is bounded by Quebec, Canada, to the north and northwest, Maine and the Atlantic Ocean to the east, Massachusetts to the south, and Vermont to the west. New Hampshires major regions are the Great North Woods, the White Mountains, the Lakes Region, the Seacoast, the Merrimack Valley, the Monadnock Region, and the Dartmouth-Lake Sunapee area. New Hampshire has the shortest ocean coastline of any U. S. coastal state, New Hampshire was home to the rock formation called the Old Man of the Mountain, a face-like profile in Franconia Notch, until the formation disintegrated in May 2003. Major rivers include the 110-mile Merrimack River, which bisects the lower half of the state north–south and ends up in Newburyport and its tributaries include the Contoocook River, Pemigewasset River, and Winnipesaukee River. The 410-mile Connecticut River, which starts at New Hampshires Connecticut Lakes and flows south to Connecticut, only one town – Pittsburg – shares a land border with the state of Vermont. The northwesternmost headwaters of the Connecticut also define the Canada–U. S, the Piscataqua River and its several tributaries form the states only significant ocean port where they flow into the Atlantic at Portsmouth. The Salmon Falls River and the Piscataqua define the southern portion of the border with Maine, the U. S. Supreme Court dismissed the case in 2002, leaving ownership of the island with Maine. New Hampshire still claims sovereignty of the base, however, the largest of New Hampshires lakes is Lake Winnipesaukee, which covers 71 square miles in the east-central part of New Hampshire. Umbagog Lake along the Maine border, approximately 12.3 square miles, is a distant second, Squam Lake is the second largest lake entirely in New Hampshire. New Hampshire has the shortest ocean coastline of any state in the United States, Hampton Beach is a popular local summer destination. It is the state with the highest percentage of area in the country. New Hampshire is in the temperate broadleaf and mixed forests biome, much of the state, in particular the White Mountains, is covered by the conifers and northern hardwoods of the New England-Acadian forests

30.
Sphenic number
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In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers. A sphenic number is a product pqr where p, q and this definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance,60 =22 ×3 ×5 has exactly 3 prime factors, the smallest sphenic number is 30 =2 ×3 ×5, the product of the smallest three primes. The first few numbers are 30,42,66,70,78,102,105,110,114,130,138,154,165. As of January 2016 the largest known number is × ×. It is the product of the three largest known primes, all sphenic numbers have exactly eight divisors. If we express the number as n = p ⋅ q ⋅ r, where p, q. For example,24 is not a number, but it has exactly eight divisors. All sphenic numbers are by definition squarefree, because the factors must be distinct. The Möbius function of any number is −1. The cyclotomic polynomials Φ n, taken over all sphenic numbers n, the first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17,1310 = 2×5×131, and 1311 = 3×19×23, there is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. The numbers 2013,2014, and 2015 are all sphenic, the next three consecutive sphenic years will be 2665,2666 and 2667. Semiprimes, products of two prime numbers

Sphenic number
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Overview

31.
Mertens function
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In number theory, the Mertens function is defined for all positive integers n as M = ∑ k =1 n μ where μ is the Möbius function. The function is named in honour of Franz Mertens and this definition can be extended to positive real numbers as follows, M = M. Less formally, M is the count of square-free integers up to x that have a number of prime factors. Because the Möbius function only takes the values −1,0, and +1, the Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture was proven false in 1985 by Andrew Odlyzko, however, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M, namely M = O. Since high values for M grow at least as fast as the root of x. Here, O refers to Big O notation, the true rate of growth of M is not known. An unpublished conjecture of Steve Gonek states that 0 < lim sup x → ∞ | M | x 5 /4 < ∞, probabilistic evidence towards this conjecture is given by Nathan Ng. Using the Euler product one finds that 1 ζ = ∏ p = ∑ n =1 ∞ μ n s where ζ is the Riemann zeta function and the product is taken over primes. Then, using this Dirichlet series with Perrons formula, one obtains,12 π i ∫ c − i ∞ c + i ∞ x s s ζ d s = M where c >1. Conversely, one has the Mellin transform 1 ζ = s ∫1 ∞ M x s +1 d x which holds for R e >1. A curious relation given by Mertens himself involving the second Chebyshev function is ψ = M log ⁡ + M log ⁡ + M log ⁡ + ⋯. Assuming that there are not multiple non-trivial roots of ζ we have the formula by the residue theorem. Weyl conjectured that the Mertens function satisfied the approximate functional-differential equation y 2 − ∑ r =1 N B2 r. Another formula for the Mertens function is M = ∑ a ∈ F n e 2 π i a where F n is the Farey sequence of order n and this formula is used in the proof of the Franel–Landau theorem. M is the determinant of the n × n Redheffer matrix, using sieve methods similar to those used in prime counting, the Mertens function has been computed for all integers up to an increasing range of x. The Mertens function for all values up to x may be computed in O time. Combinatorial based algorithms can compute isolated values of M in O time, see A084237 for values of M at powers of 10

Mertens function
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Mertens function to n=10,000

32.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently

Fibonacci number
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A page of Fibonacci 's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.
Fibonacci number
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A tiling with squares whose side lengths are successive Fibonacci numbers

33.
Markov number
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The first few Markov numbers are 1,2,5,13,29,34,89,169,194,233,433,610,985,1325. Appearing as coordinates of the Markov triples, etc, there are infinitely many Markov numbers and Markov triples. There are two ways to obtain a new Markov triple from an old one. First, one may permute the 3 numbers x, y, z, second, if is a Markov triple then by Vieta jumping so is. Applying this operation twice returns the same triple one started with, joining each normalized Markov triple to the 1,2, or 3 normalized triples one can obtain from this gives a graph starting from as in the diagram. This graph is connected, in other words every Markov triple can be connected to by a sequence of these operations. If we start, as an example, with we get its three neighbors, and in the Markov tree if x is set to 1,5 and 13, respectively. For instance, starting with and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers, starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers. All the Markov numbers on the adjacent to 2s region are odd-indexed Pell numbers. Thus, there are infinitely many Markov triples of the form, likewise, there are infinitely many Markov triples of the form, where Px is the xth Pell number. Aside from the two smallest singular triples and, every Markov triple consists of three distinct integers, odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32. In his 1982 paper, Don Zagier conjectured that the nth Markov number is given by m n =13 e C n + o with C =2.3523414972 …. Moreover, he pointed out that x 2 + y 2 + z 2 =3 x y z +4 /9, the conjecture was proved by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry. The nth Lagrange number can be calculated from the nth Markov number with the formula L n =9 −4 m n 2, the Markov numbers are sums of pairs of squares. If X⋅Y⋅Z =1 then Tr = Tr, so more symmetrically if X, Y, and Z are in SL2 with X⋅Y⋅Z =1, cambridge Tracts in Mathematics and Mathematical Physics. Markov spectrum problem, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Markoff, A. Sur les formes quadratiques binaires indéfinies

Markov number
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The first levels of the Markov number tree

34.
610 (telephone)
–
A600 series connector is an obsolete three-pin connector with up to six conductors. It was for years the standard telephone service connector in Australia. As of 2008 a large installed base remained and a variety of connectors, adapters and equipment such as ADSL filters were still in production. It has no common or standard use, either within Australia or beyond. However it is manufactured in countries, usually in China. The 600 series plug has up to three flat pins plus a non-conducting spigot, each pin carries up to two conductors by means of two contacts, one on each side of the pin. The pin bodies are of non-conductive material, the original series 600 plugs and sockets were designed to be equally suitable for fixed wall mounting or for use on flexible cords. Such connectors are available, but specialised fixed and line versions also exist. Often now replaced by a modular 6P2C connector and an adaptor, In particular, the non-conducting spigot of the original 605 plug was provided with a hole for a countersunk screw, allowing the plug to be fixed in a wall-mounted socket. Removal then required first removing the cover, to allow this retaining screw to be removed. In this way a semi-permanent connection, requiring tools for disconnection, the original 610 socket was provided with two mounting holes, one behind the receptacle for the plug spigot, and both for countersunk screws. Similar to 610, but with the contacts of pairs one and this is particularly designed for mode 3 connection. The incoming line to the mode 3 device is connected using pair one, if the mode 3 device is unplugged, the switch contacts maintain line connection to the other devices. This function is not generally supported by modular connectors, modular connectors providing suitable switch contacts are available but not common, Instead, mode 3 devices use two modular connectors, one for the incoming line and the other for the outgoing line. The convenience of using a connector in this application is one reason for the continued use of 600 series connectors. For example, if a modern modem with its supplied 6P2C adaptor is plugged into a conventionally wired 611 mode 3 socket, in some sockets conductors 3 and 4 also connect when no plug is present, shorting pair three. Sockets are also available with cams to allow the switching functions to be enabled as required. In some older sockets this could be achieved by bending the contacts

610 (telephone)
–
600 series connectors

35.
Australia
–
Australia, officially the Commonwealth of Australia, is a country comprising the mainland of the Australian continent, the island of Tasmania and numerous smaller islands. It is the worlds sixth-largest country by total area, the neighbouring countries are Papua New Guinea, Indonesia and East Timor to the north, the Solomon Islands and Vanuatu to the north-east, and New Zealand to the south-east. Australias capital is Canberra, and its largest urban area is Sydney, for about 50,000 years before the first British settlement in the late 18th century, Australia was inhabited by indigenous Australians, who spoke languages classifiable into roughly 250 groups. The population grew steadily in subsequent decades, and by the 1850s most of the continent had been explored, on 1 January 1901, the six colonies federated, forming the Commonwealth of Australia. Australia has since maintained a liberal democratic political system that functions as a federal parliamentary constitutional monarchy comprising six states. The population of 24 million is highly urbanised and heavily concentrated on the eastern seaboard, Australia has the worlds 13th-largest economy and ninth-highest per capita income. With the second-highest human development index globally, the country highly in quality of life, health, education, economic freedom. The name Australia is derived from the Latin Terra Australis a name used for putative lands in the southern hemisphere since ancient times, the Dutch adjectival form Australische was used in a Dutch book in Batavia in 1638, to refer to the newly discovered lands to the south. On 12 December 1817, Macquarie recommended to the Colonial Office that it be formally adopted, in 1824, the Admiralty agreed that the continent should be known officially as Australia. The first official published use of the term Australia came with the 1830 publication of The Australia Directory and these first inhabitants may have been ancestors of modern Indigenous Australians. The Torres Strait Islanders, ethnically Melanesian, were originally horticulturists, the northern coasts and waters of Australia were visited sporadically by fishermen from Maritime Southeast Asia. The first recorded European sighting of the Australian mainland, and the first recorded European landfall on the Australian continent, are attributed to the Dutch. The first ship and crew to chart the Australian coast and meet with Aboriginal people was the Duyfken captained by Dutch navigator, Willem Janszoon. He sighted the coast of Cape York Peninsula in early 1606, the Dutch charted the whole of the western and northern coastlines and named the island continent New Holland during the 17th century, but made no attempt at settlement. William Dampier, an English explorer and privateer, landed on the north-west coast of New Holland in 1688, in 1770, James Cook sailed along and mapped the east coast, which he named New South Wales and claimed for Great Britain. The first settlement led to the foundation of Sydney, and the exploration, a British settlement was established in Van Diemens Land, now known as Tasmania, in 1803, and it became a separate colony in 1825. The United Kingdom formally claimed the part of Western Australia in 1828. Separate colonies were carved from parts of New South Wales, South Australia in 1836, Victoria in 1851, the Northern Territory was founded in 1911 when it was excised from South Australia

Australia
–
Aboriginal rock art in the Kimberley region of Western Australia
Australia
Australia
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Portrait of Captain James Cook, the first European to map the eastern coastline of Australia in 1770
Australia
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Tasmania's Port Arthur penal settlement is one of eleven UNESCO World Heritage-listed Australian Convict Sites.

36.
Area code 612
–
By geographical area, it is the smallest area code in the state of Minnesota. However, like every single metropolitan area code in the United States, the used to be much larger, accounting for the entire Twin Cities region. At the outset, Minnesota received two area codes,612 and 218, a 1947 map of the NANP showed the region defined as roughly the southeastern third of Minnesota. The rest of the state was 218, which formed a region around the 612 area code. The separating line extended westward from Duluth to the center of the state, in 1954, the state was divided into three area codes. Part of the portion of the previous 612 territory, including Rochester. The 612 area code was rotated out to reach the edge of the state. The 218 region was reshaped to be square, absorbing much of the old 612s northeastern portion. This configuration remained in place for 42 years, in 1996, almost all of the old 612 territory outside of the Twin Cities became area code 320. The 612 region was split in half two years later in 1998, mostly following the Mississippi River. The area west of the Mississippi, including Minneapolis, retained the old code and this was intended to be a long-term solution for exchanges in the Twin Cities. However, the Twin Cities are not only home to most of the states landlines, as a result,612 was on the verge of exhaustion again within less than a year of the 651 split. As a result, the 612 code shrank to its current size in a split that took effect in 2000. Area code 763 was created to include the northwest suburbs, the area code splits in the Twin Cities are unusual because they split along municipal, rather than central office, boundaries. This led to a number of exchanges being divided between two area codes, and a few being divided among three. The eastern half of the University of Minnesota, Twin Cities campus, because of an integrated phone system linking both campuses, the Falcon Heights campus remained in 612 after the 1998 split. Portions of area codes 320 and 507 are local calls from the Twin Cities as well, city of Minneapolis Richfield St. Anthony Fort Snelling University of Minnesota List of North American area codes NANPA, Minnesota area code map Area code history 1947 Area Code Assignment Map

Area code 612
–
Metropolitan Minneapolis area codes with 612 in yellow.

37.
Judaism
–
Judaism encompasses the religion, philosophy, culture and way of life of the Jewish people. Judaism is an ancient monotheistic Abrahamic religion, with the Torah as its text, and supplemental oral tradition represented by later texts such as the Midrash. Judaism is considered by religious Jews to be the expression of the relationship that God established with the Children of Israel. With between 14.5 and 17.4 million adherents worldwide, Judaism is the tenth-largest religion in the world, Judaism includes a wide corpus of texts, practices, theological positions, and forms of organization. Modern branches of Judaism such as Humanistic Judaism may be nontheistic, today, the largest Jewish religious movements are Orthodox Judaism, Conservative Judaism and Reform Judaism. Major sources of difference between groups are their approaches to Jewish law, the authority of the Rabbinic tradition. Orthodox Judaism maintains that the Torah and Jewish law are divine in origin, eternal and unalterable, Conservative and Reform Judaism are more liberal, with Conservative Judaism generally promoting a more traditional interpretation of Judaisms requirements than Reform Judaism. A typical Reform position is that Jewish law should be viewed as a set of guidelines rather than as a set of restrictions and obligations whose observance is required of all Jews. Historically, special courts enforced Jewish law, today, these still exist. Authority on theological and legal matters is not vested in any one person or organization, the history of Judaism spans more than 3,000 years. Judaism has its roots as a religion in the Middle East during the Bronze Age. Judaism is considered one of the oldest monotheistic religions, the Hebrews and Israelites were already referred to as Jews in later books of the Tanakh such as the Book of Esther, with the term Jews replacing the title Children of Israel. Judaisms texts, traditions and values strongly influenced later Abrahamic religions, including Christianity, Islam, many aspects of Judaism have also directly or indirectly influenced secular Western ethics and civil law. Jews are a group and include those born Jewish and converts to Judaism. In 2015, the world Jewish population was estimated at about 14.3 million, Judaism thus begins with ethical monotheism, the belief that God is one and is concerned with the actions of humankind. According to the Tanakh, God promised Abraham to make of his offspring a great nation, many generations later, he commanded the nation of Israel to love and worship only one God, that is, the Jewish nation is to reciprocate Gods concern for the world. He also commanded the Jewish people to one another, that is. These commandments are but two of a corpus of commandments and laws that constitute this covenant, which is the substance of Judaism

38.
Kabbalah
–
Kabbalah is an esoteric method, discipline, and school of thought that originated in Judaism. A traditional Kabbalist in Judaism is called a Mekubbal, Kabbalah is a set of esoteric teachings meant to explain the relationship between an unchanging, eternal, and mysterious Ein Sof and the mortal and finite universe. While it is used by some denominations, it is not a religious denomination in itself. It forms the foundations of religious interpretation. Kabbalah seeks to define the nature of the universe and the human being, the nature and purpose of existence and it also presents methods to aid understanding of the concepts and thereby attain spiritual realisation. Kabbalah originally developed within the realm of Jewish tradition, and kabbalists often use classical Jewish sources to explain, traditional practitioners believe its earliest origins pre-date world religions, forming the primordial blueprint for Creations philosophies, religions, sciences, arts, and political systems. Safed Rabbi Isaac Luria is considered the father of contemporary Kabbalah and it was popularised in the form of Hasidic Judaism from the 18th century onwards. According to the Zohar, a text for kabbalistic thought. These four levels are called pardes from their initial letters, peshat, the direct interpretations of meaning. Derash, midrashic meanings, often with imaginative comparisons with similar words or verses, sod, the inner, esoteric meanings, expressed in kabbalah. Kabbalah is considered by its followers as a part of the study of Torah – the study of Torah being an inherent duty of observant Jews. A third tradition, related but more shunned, involves the magical aims of Practical Kabbalah and they can be readily distinguished by their basic intent with respect to God, The Theosophical tradition of Theoretical Kabbalah seeks to understand and describe the divine realm. Consequently, it formed a minor tradition shunned from Kabbalah. According to traditional belief, early kabbalistic knowledge was transmitted orally by the Patriarchs, prophets, According to this view, early kabbalah was, in around the 10th century BC, an open knowledge practiced by over a million people in ancient Israel. Foreign conquests drove the Jewish spiritual leadership of the time to hide the knowledge and make it secret and it is hard to clarify with any degree of certainty the exact concepts within kabbalah. There are several different schools of thought with different outlooks, however. From the Renaissance onwards Jewish Kabbalah texts entered non-Jewish culture, where they were studied and translated by Christian Hebraists, syncretic traditions of Christian Kabbalah and Hermetic Qabalah developed independently of Jewish Kabbalah, reading the Jewish texts as universal ancient wisdom. Both adapted the Jewish concepts freely from their Judaic understanding, to merge with other theologies, religious traditions, with the decline of Christian Cabala in the Age of Reason, Hermetic Qabalah continued as a central underground tradition in Western esotericism

39.
Sefirah
–
The term is alternatively transliterated into English as Sefirot/Sefiroth, singular Sephirah/Sefirah etc. Alternative configurations of the sephirot are given by different schools in the development of Kabbalah. The tradition of enumerating 10 is stated in the Sefer Yetzirah, Ten sephirot of nothingness, ten and not nine, ten and not eleven. As altogether 11 sephirot are listed across the different schemes, two are seen as unconscious and conscious manifestations of the principle, conserving the ten categories. In Kabbalah the functional structure of the sephirot in channeling Divine creative life force, the first sephirah describes the Divine Will above intellect. The next sephirot describe conscious Divine Intellect, and the latter sephirot describe the primary and secondary conscious Divine Emotions, two sephirot are feminine, as the female principle in Kabbalah describes a vessel that receives the outward male light, then inwardly nurtures and gives birth to lower sephirot. Corresponding to this is the Female Divine Presence, Kabbalah sees the human soul as mirroring the Divine, and more widely, all creations as reflections of their life source in the sephirot. Therefore, the sephirot also describe the life of man. This relationship between the soul of man and the Divine, gives Kabbalah one of its two central metaphors in describing Divinity, alongside the other Ohr metaphor, however, Kabbalah repeatedly stresses the need to avoid all corporeal interpretation. Through this, the sephirot are related to the structure of the body and are reformed into Partsufim, underlying the structural purpose of each sephirah is a hidden motivational force which is understood best by comparison with a corresponding psychological state in human spiritual experience. The term sefirah thus has complex connotations within Kabbalah, the Sephirot are considered revelations of the Creators Will, and they should not be understood as ten different gods but as ten different ways the one God reveals his Will through the Emanations. While in Cordoveran Kabbalah, Keter is listed as the first Sephirah, the Sephirot are emanated from the Divine Will, because Kabbalah sees different levels within Keter, reflecting Gods inner Will and outer Will. The innermost, hidden levels of Keter, also in some contexts called The head/beginning that is not known, are united above the Sephirot with the Ein Sof and it is not God who changes but the ability to perceive God that changes. This difference between the MaOhr and the Ohr He emanates is stressed in Kabbalah, so as to avoid notions of any plurality in the Godhead. In its early 12th-century dissemination, Kabbalah received criticism from some Rabbis, the multiplicity of revealed emanations only applies from the perspective of the Creation, and not from the perspective of the infinite Divine essence. The ten Sephirot are a step-by-step process illuminating the Divine plan as it unfolds itself in Creation and they are fully found in the Medieval Kabbalah texts, such as the central work in Kabbalah, the Zohar. The Hebrew etymology of their names in Kabbalah is understood to refer to the aspects of meaning of each Sephirah. Kabbalah expounds on the terms of the Sephirot and this difference of opinion reflects earlier Medieval debate on whether Keter can be identified with the Ohr Ein Sof itself, or as the first revealed Sephirah

Sefirah
–
The Yosher -Upright configuration of the 10 Sephirot, arranged into 3 columns
Sefirah
–
Kabbalah relates the Sephirot and Indwelling Shechinah Presence to Male-Female Divine principles, represented in the union of Jewish marriage Below. In Medieval Kabbalah the task of man is Yichud-"Union" of Male and Female Divinity on High [citation needed]. In Lurianic Kabbalah man redeems exiled Sparks of Holiness of the Shechinah from material Kelipot
Sefirah
–
Ten Sephirot on the east wall of the Cordoba Synagogue, Spain

40.
613 mitzvot
–
The tradition that 613 commandments is the number of mitzvot in the Torah, began in the 3rd century CE, when Rabbi Simlai mentioned it in a sermon that is recorded in Talmud Makkot 23b. These principles of Biblical law are sometimes called connections or commandments and referred to collectively as the Law of Moses, Mosaic Law, Sinaitic Law, the word mitzvot is plural, singular is mitzvah. Although there have been attempts to codify and enumerate the commandments contained in the Torah. The 613 commandments include positive commandments, to perform an act, though the number 613 is mentioned in the Talmud, its real significance increased in later medieval rabbinic literature, including many works listing or arranged by the mitzvot. Three types of negative commandments fall under the self-sacrificial principle yehareg veal yaavor and these are murder, idolatry, and forbidden sexual relations. The 613 mitzvot have been divided also into three categories, mishpatim, edot, and chukim. Mishpatim include commandments that are deemed to be self-evident, such as not to murder, edot commemorate important events in Jewish history. For example, the Shabbat is said to testify to the story that Hashem created the world in six days and rested on the seventh day, chukim are commandments with no known rationale, and are perceived as pure manifestations of the Divine will. Many of the mitzvot cannot be observed now, following the destruction of the Second Temple, According to one standard reckoning, there are 77 positive and 194 negative commandments that can be observed today, of which there are 26 commands that apply only within the Land of Israel. Furthermore, there are some time-related commandments from which women are exempt, some depend on the special status of a person in Judaism, while others apply only to men or only to women. 33,04 is to be interpreted to mean that Moses transmitted the Torah from God to the Israelites, Moses commanded us the Torah as an inheritance for the community of Jacob. The Talmud attributes the number 613 to Rabbi Simlai, but other classical sages who hold this view include Rabbi Simeon ben Azzai and it is quoted in Midrash Shemot Rabbah 33,7, Bamidbar Rabbah 13, 15–16,18,21 and Talmud Yevamot 47b. Many Jewish philosophical and mystical works find allusions and inspirational calculations relating to the number of commandments. The tzitzit of the tallit are connected to the 613 commandments by interpretation, principal Torah commentator Rashi bases the number of knots on a gematria, Each tassel has eight threads and five sets of knots, totalling 13. The sum of all numbers is 613 and this reflects the concept that donning a garment with tzitzit reminds its wearer of all Torah commandments. Rabbinic support for the number of commandments being 613 is not without dissent and, even as the number gained acceptance, some rabbis declared that this count was not an authentic tradition, or that it was not logically possible to come up with a systematic count. No early work of Jewish law or Biblical commentary depended on the 613 system, the classical Biblical commentator and grammarian Rabbi Abraham ibn Ezra denied that this was an authentic rabbinic tradition. Nahmanides held that this particular counting was a matter of controversy

41.
Torah
–
The Torah is the central reference of Judaism. It has a range of meanings and it can most specifically mean the first five books of the twenty-four books of the Tanakh, and it usually includes the rabbinic commentaries. In rabbinic literature the word Torah denotes both the five books and the Oral Torah, the Oral Torah consists of interpretations and amplifications which according to rabbinic tradition have been handed down from generation to generation and are now embodied in the Talmud and Midrash. According to the Midrash, the Torah was created prior to the creation of the world, traditionally, the words of the Torah are written on a scroll by a scribe in Hebrew. A Torah portion is read publicly at least once every three days in the presence of a congregation, reading the Torah publicly is one of the bases for Jewish communal life. The word Torah in Hebrew is derived from the root ירה, the meaning of the word is therefore teaching, doctrine, or instruction, the commonly accepted law gives a wrong impression. Other translational contexts in the English language include custom, theory, guidance, the earliest name for the first part of the Bible seems to have been The Torah of Moses. This title, however, is neither in the Torah itself. It appears in Joshua and Kings, but it cannot be said to refer there to the entire corpus, in contrast, there is every likelihood that its use in the post-Exilic works was intended to be comprehensive. Other early titles were The Book of Moses and The Book of the Torah, Christian scholars usually refer to the first five books of the Hebrew Bible as the Pentateuch, a term first used in the Hellenistic Judaism of Alexandria, meaning five books, or as the Law. The Torah starts from the beginning of Gods creating the world, through the beginnings of the people of Israel, their descent into Egypt, and it ends with the death of Moses, just before the people of Israel cross to the promised land of Canaan. Interspersed in the narrative are the teachings given explicitly or implicitly embedded in the narrative. This is followed by the story of the three patriarchs, Joseph and the four matriarchs, God gives to the patriarchs a promise of the land of Canaan, but at the end of Genesis the sons of Jacob end up leaving Canaan for Egypt due to a regional famine. They had heard there was a grain storage and distribution facility in Egypt. Exodus begins the story of Gods revelation to his people of Israel through Moses, Moses receives the Torah from God, and teaches His laws and Covenant to the people of Israel. It also talks about the first violation of the covenant when the Golden Calf was constructed, Exodus includes the instructions on building the Tabernacle and concludes with its actual construction. Leviticus begins with instructions to the Israelites on how to use the Tabernacle, leviticus 26 provides a detailed list of rewards for following Gods commandments and a detailed list of punishments for not following them. Numbers tells how Israel consolidated itself as a community at Sinai, set out from Sinai to move towards Canaan, even Moses sins and is told he would not live to enter the land

42.
Madison Square Garden
–
Madison Square Garden, often called MSG or simply The Garden, is a multi-purpose indoor arena in the New York City borough of Manhattan. Located in Midtown Manhattan between 7th and 8th Avenues from 31st to 33rd Streets, it is situated atop Pennsylvania Station. The Garden is used for basketball and ice hockey, as well as boxing, concerts, ice shows, circuses, professional wrestling and other forms of sports. It is close to other midtown Manhattan landmarks, including the Empire State Building, Koreatown and it is home to the New York Rangers of the National Hockey League, the New York Knicks of the National Basketball Association, and residency to singer-songwriter Billy Joel. The Garden opened on February 11,1968, and is the oldest major sporting facility in the New York metropolitan area and it is the oldest arena in the National Hockey League and the second-oldest arena in the National Basketball Association. MSG is the fourth-busiest music arena in the world in terms of sales, behind The O2 Arena. At a total construction cost of approximately $1.1 billion and it is part of the Pennsylvania Plaza office and retail complex. Several other operating entities related to the Garden share its name, Madison Square is formed by the intersection of 5th Avenue and Broadway at 23rd Street in Manhattan. It was named after James Madison, fourth President of the United States, two venues called Madison Square Garden were located just northeast of the square, the first from 1879 to 1890, and the second from 1890 to 1925. The first Garden, leased to P. T. Barnum, had no roof and was inconvenient to use during inclement weather, Madison Square Garden II was designed by noted architect Stanford White. The new building was built by a syndicate which included J. P. Morgan, Andrew Carnegie, P. T. Barnum, Darius Mills, James Stillman and W. W. Astor. It was 200 feet by 485 feet, and the main hall and it had a 1, 200-seat theatre, a concert hall with a capacity of 1,500, the largest restaurant in the city and a roof garden cabaret. A third Madison Square Garden opened in a new location, on 8th Avenue between 49th and 50th Streets, from 1925 to 1968, groundbreaking on the third Madison Square Garden took place on January 9,1925. Designed by the theater architect Thomas W. Lamb, it was built at the cost of $4.75 million in 249 days by boxing promoter Tex Rickard. The arena was 200 feet by 375 feet, with seating on three levels, and a capacity of 18,496 spectators for boxing. Demolition commenced in 1968 after the opening of the current Garden and it finished up in early 1969, and the site is now where One Worldwide Plaza is located. The new structure was one of the first of its kind to be built above the platforms of a railroad station. It was an engineering feat constructed by Robert E. McKee of El Paso, public outcry over the demolition of the Pennsylvania Station structure—an outstanding example of Beaux-Arts architecture—led to the creation of the New York City Landmarks Preservation Commission

Madison Square Garden
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The Eighth Avenue facade of Madison Square Garden in August 2009
Madison Square Garden
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A basketball game at Madison Square Garden circa 1968
Madison Square Garden
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Madison Square Garden's upper bowl concourse, seen in January 2014 during a Rangers game.
Madison Square Garden
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The completely transformed Madison Square Garden in January 2014 (with a new HD scoreboard), as the New York Rangers play against the St. Louis Blues.

43.
New York Knicks
–
The New York Knickerbockers, commonly referred to as the Knicks, are an American professional basketball team based in New York City. The Knicks compete in the National Basketball Association as a club of the Atlantic Division of the Eastern Conference. The team plays its games at Madison Square Garden, located in the borough of Manhattan. They are one of two NBA teams located in New York City, the other is the Brooklyn Nets, along with the Boston Celtics, the Knicks are one of only two original NBA teams still located in its original city. The Knicks were successful during their years and were constant playoff contenders under the franchises first head coach Joe Lapchick. Beginning in 1950, the Knicks made three appearances in the NBA Finals, all of which were losing efforts. Lapchick resigned in 1956 and the team began to falter. It was not until the late 1960s when Red Holzman became head coach that the Knicks began to regain their former dominance, Holzman successfully guided the Knicks to two NBA championships, in 1970 and 1973. The Knicks of the 1980s had mixed success that included six playoff appearances, however, the playoff-level Knicks of the 1990s were led by future Hall of Fame center Patrick Ewing, this era was marked by passionate rivalries with the Chicago Bulls, Indiana Pacers, and Miami Heat. During this time, they were known for playing tough defense under head coaches Pat Riley, during this era, the Knicks made two appearances in the NBA Finals, in 1994 and 1999, though they were unable to win an NBA championship. Since 2000, the Knicks have struggled to regain their former glory, in 2012–13, the franchise won its first division title in 19 years, but was eliminated in the second round of the playoffs by the Indiana Pacers. According to a 2016 Forbes report, the Knicks were the most-valuable NBA franchise, in 1946, basketball, particularly college basketball, was a growing and increasingly profitable sport in New York City. Hockey was another sport at the time and generated considerable profits, however. Max Kase, a New York sportswriter, became the editor at the Boston American in the 1930s. Kase developed the idea of a professional league to showcase college players upon their graduation. Brown, intrigued by the opportunity to attain additional income when the teams were not playing or on the road. Ned Irish, a college basketball promoter, retired sportswriter and then president of Madison Square Garden, was in attendance, Kase originally planned to own and operate the New York franchise himself and approached Irish with a proposal to lease the Garden. Irish explained that the rules of the Arena Managers Association of America stated that Madison Square Garden was required to own any professional teams played in the arena

New York Knicks
–
Lapchick was responsible for leading the Knicks during their early success. However, these ventures never culminated with a win in the NBA Finals.
New York Knicks
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New York Knicks
New York Knicks
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William 'Red' Holzman guided the Knicks to two championships during his tenure.
New York Knicks
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Walt 'Clyde' Frazier

44.
Red Holzman
–
William Red Holzman was an NBA basketball player and coach probably best known as the head coach of the New York Knicks from 1967 to 1982. Holzman helped lead the Knicks to two NBA Championships in 1970 and 1973, and was inducted into the Basketball Hall of Fame in 1986, in 1996, Holzman was named one of Top 10 Coaches in NBA History. Born in Brooklyn, New York in 1920, to Jewish immigrant parents, as the son of a Romanian mother, Holzman grew up in that boroughs Ocean Hill–Brownsville neighborhood and played basketball for Franklin K. Lane High School in the mid-1930s. He attended the University of Baltimore and later the City College of New York, Holzman joined the United States Navy in the same year, and played on the Norfolk, Virginia Naval Base team for two years. Holzman was discharged from the Navy in 1945 and subsequently joined the NBL Rochester Royals, Holzman was Rookie of the Year in 1944–45. In 1945–46 and 1947–48 he was on the NBLs first All League team, Holzman stayed with the team through their move to the NBA and subsequent NBA championship in 1951. In 1953, Holzman left the Royals and joined the Milwaukee Hawks as a player-coach, eventually retiring as a player in 1954, during the 1956–1957 season, Holzman led the Hawks to 19 losses during their first 33 games, and was subsequently fired. In 1957, Holzman became a scout for the New York Knicks for ten years ending in 1967, during this 15-year span as Knicks coach, Holzman won a total of 613 games, including two NBA championships in 1970 and 1973. In 1969, Holzman coached the Knicks to a then single-season NBA record 18-game win streak, for his efforts leading up to the Knicks 1970 championship win, Holzman was named the NBA Coach of the Year for that year. He was one of very few individuals to have won an NBA championship as both player and coach, as a coach, his final record was 696 wins and 604 losses. In 1985, he was elected into the Naismith Memorial Basketball Hall of Fame, the New York Knicks have retired the number 613 in his honor, equaling the number of wins he accumulated as their head coach. He lived with his wife in a home they bought in Cedarhurst, following his lengthy NBA coaching career, Holzman was diagnosed with leukemia and died at Long Island Jewish Medical Center in New Hyde Park, New York in 1998

Red Holzman
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Red Holzman in the 1970s
Red Holzman
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Holzman in 1950, when playing for the Rochester Royals.

45.
Padovan sequence
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The Padovan sequence is the sequence of integers P defined by the initial values P = P = P =1, and the recurrence relation P = P + P. The first few values of P are 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265. The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom, Hans van der Laan, Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996 and he also writes about it in one of his books, Math Hysteria, Fun Games With Mathematics. The above definition is the one given by Ian Stewart and by MathWorld, other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets. This is a property of recurrence relations, the Perrin sequence can be obtained from the Padovan sequence by the following formula, P e r r i n = P + P. e. The Padovan sequence also satisfies the identity P2 − P P = P. The Padovan sequence is related to sums of binomial coefficients by the following identity, P = ∑2 m + n = k = ∑ m = ⌈ k /3 ⌉ ⌊ k /2 ⌋. For example, for k =12, the values for the pair with 2m + n =12 which give non-zero binomial coefficients are, and, and, + + =1 +10 +1 =12 = P. The Padovan sequence numbers can be written in terms of powers of the roots of the equation x 3 − x −1 =0 and this equation has 3 roots, one real root p and two complex conjugate roots q and r. Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r, P = a p n + b q n + c r n where a, b and c are constants. Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n, and P − a p n tends to zero. For all n ≥0, P is the integer closest to p n −1 s, the ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the ratio does to the Fibonacci sequence. P is the number of ways of writing n +2 as a sum in which each term is either 2 or 3. This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as, ∑ n =0 ∞ P α n = α2 α3 − α −1. A Padovan prime is P that is prime, the first few Padovan primes are 2,3,5,7,37,151,3329,23833. Also, if you count the number of As, Bs and Cs in each string, then for the nth string, you have P As, P Bs, the count of BB pairs, AA pairs and CC pairs are also Padovan numbers

Padovan sequence
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Spiral of equilateral triangles with side lengths which follow the Padovan sequence.

46.
Number of the Beast (numerology)
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The Number of the Beast is a term in the Book of Revelation, of the New Testament, that is associated with the Beast of Revelation in chapter 13. In most manuscripts of the New Testament and in English translations of the Bible, in critical editions of the Greek text, such as the Novum Testamentum Graece, it is noted that 616 is a variant. In the oldest preserved manuscript as of 2017, as well as ancient sources like Codex Ephraemi Rescriptus. The Number of the beast is described in the passage of Revelation 13, the actual number is only mentioned once, in verse 18. Possible translations include not only to count, to reckon but also to vote or to decide, in the Textus Receptus, derived from Byzantine text-type manuscripts, the number 666 is represented by the final 3 letters χξϛ. 17καὶ ἵνα μή τις δύνηται ἀγοράσαι ἢ πωλῆσαι εἰ μὴ ὁ ἔχων τὸ χάραγμα, 18Ὧδε ἡ σοφία ἐστίν· ὁ ἔχων τὸν νοῦν ψηφισάτω τὸν ἀριθμὸν τοῦ θηρίου· ἀριθμὸς γὰρ ἀνθρώπου ἐστί· καὶ ὁ ἀριθμὸς αὐτοῦ χξϛʹ. The last letter of the Greek alphabet is not the equivalent of the English letter Z, the Greek letter stigma ligature represents the number 6. 18ὧδε ἡ σοφία ἐστίν· ὁ ἔχων νοῦν ψηφισάτω τὸν ἀριθμὸν τοῦ θηρίου, irenaeus knew about the 616 reading, but did not adopt it. In the 380s, correcting the existing Latin-language version of the New Testament, around 2005, a fragment from Papyrus 115, taken from the Oxyrhynchus site, was discovered at the Oxford Universitys Ashmolean Museum. It gave the number as 616 χιϛʹ. This fragment is the oldest manuscript of Revelation 13 found as of 2017, the age of a manuscript is not an indicator of the date of its writing but refers to how old the physical material is. All original biblical manuscripts are non-existent today, as they were held and copied onto new materials, eventually the originals fell apart, leaving fragments for a period and then only the copies. So the oldest texts might actually be found among the newest copies, Codex Ephraemi Rescriptus, known before the P115 finding but dating to after it, has 616 written in full, ἑξακόσιοι δέκα ἕξ, hexakosioi deka hex. Papyrus 115 and Ephraemi Rescriptus have led scholars to regard 616 as the original number of the beast. Associating the number of the beast as the duration of the beast’s reign Corresponding symbolism for the Antichrist, in Greek isopsephy and Hebrew gematria, every letter has a corresponding numeric value. Summing these numbers gives a value to a word or name. The use of isopsephy to calculate the number of the beast is used in many of the below interpretations, preterist theologians typically support the numerical interpretation that 666 is the equivalent of the name and title, Nero Caesar. A manner of speaking against the emperor without the Roman authorities knowing, also Nero Caesar in the Hebrew alphabet is נרון קסר NRON QSR, which when used as numbers represent 5020065010060200, which add to 666

Number of the Beast (numerology)
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The number of the beast is 666 by William Blake.
Number of the Beast (numerology)
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Fragment from Papyrus 115 (P115) of Revelation in the 66th vol. of the Oxyrhynchus series (P. Oxy. 4499). Has the number of the beast as χιϛ, 616.
Number of the Beast (numerology)
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Bust of Nero at Musei Capitolini, Rome

47.
666 (number)
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666 is the natural number following 665 and preceding 667. Six hundred and sixty-six is called the number of the Beast in chapter 13 of the Book of Revelation, of the New Testament,666 is the sum of the first 36 natural numbers, and thus it is a triangular number. Notice that 36 =15 +21,15 and 21 are also triangular numbers, in base 10,666 is a repdigit and a Smith number. A prime reciprocal magic square based on 1/149 in base 10 has a total of 666. The prime factorization of 666 is 2 •32 •37, some manuscripts of the original Greek use the symbols χξϛ chi xi stigma, while other manuscripts spell out the number in words. In modern popular culture,666 has become one of the most widely recognized symbols for the Antichrist or, alternatively, the number 666 is purportedly used to invoke Satan. Earnest references to the number occur both among apocalypticist Christian groups and in explicitly anti-Christian subcultures, references in contemporary Western art or literature are, more likely than not, intentional references to the Beast symbolism. Such popular references are therefore too numerous to list and it is common to see the symbolic role of the integer 666 transferred to the digit sequence 6-6-6. Some people take the Satanic associations of 666 so seriously that they actively avoid things related to 666 or the digits 6-6-6, in some early biblical manuscripts, including Papyrus 115, the number is cited as 616. In the Bible,666 is the number of talents of gold Solomon collected each year, in the Bible,666 is the number of Adonikams descendants who return to Jerusalem and Judah from the Babylonian exile. In the Bible, there may be a latent reference to 666 in the name of the great sixth-century BC king of Babylon, commonly spelled Nebuchadnezzar, transliterating from the Book of Daniel, the name is Nebuchadrezzar or Nebuchadrezzur in the Book of Jeremiah. The number of name can be calculated, since Hebrew letters double as numbers. Nebuchadrezzar is 663, and Nebuchadrezzur,669, midway between the two variants is 666. If the mysteries of Jeremiah are to be related to those of Revelation, Nebuchadrezzar, using gematria, Neron Caesar transliterated from Greek into Hebrew produces the number 666. The Latin spelling of Nero Caesar transliterated into Hebrew produces the number 616, thus, in the Bible,666 may have been a coded reference to Nero the Roman Emperor from 55 to 68 AD. Is the magic sum, or sum of the constants of a six by six magic square. Is the sum of all the numbers on a roulette wheel, was a winning lottery number in the 1980 Pennsylvania Lottery scandal, in which equipment was tampered to favor a 4 or 6 as each of the three individual random digits. Was the original name of the Macintosh SevenDust computer virus that was discovered in 1998, the number is a frequent visual element of Aryan Brotherhood tattoos

666 (number)
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666 is often associated with the devil.
666 (number)
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666 float in a Paris parade

48.
Eisenstein prime
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In mathematics, an Eisenstein prime is an Eisenstein integer z = a + b ω that is irreducible in the ring-theoretic sense, its only Eisenstein divisors are the units, a + bω itself and its associates. The associates and the conjugate of any Eisenstein prime are also prime. It follows that the absolute value squared of every Eisenstein prime is a prime or the square of a natural prime. The first few Eisenstein primes that equal a natural prime 3n −1 are,2,5,11,17,23,29,41,47,53,59,71,83,89,101. Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes, some non-real Eisenstein primes are 2 + ω,3 + ω,4 + ω,5 + 2ω,6 + ω,7 + ω,7 + 3ω. Up to conjugacy and unit multiples, the primes listed above, as of March 2017, the largest known Eisenstein prime is the seventh largest known prime 10223 ×231172165 +1, discovered by Péter Szabolcs and PrimeGrid. All larger known primes are Mersenne primes, discovered by GIMPS, real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes are congruent to 1 mod 3, thus no Mersenne prime is an Eisenstein prime

Eisenstein prime
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Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3 n − 1. All others have an absolute value squared equal to a natural prime.